The question at hand is to use Parseval’s theorem to solve the following integral: $$\int_{-\infty}^{\infty} sinc^4 (kt) dt$$ I understand Parseval’s theorem to be: $$E_g = \int_{-\infty}^{\infty} g^2(t) = \int_{-\infty}^{\infty} |G(f)|^2 df $$ I began by doing the obvious and removing the squared such that: $$g^2(t) = Sinc^4 (kt)$$ $$g(t) = Sinc^2 (kt)$$ Following the […]

I’m trying to compute the Fourier transform of $$f(\mathbf{r}) = \frac{1}{r^\alpha}$$ where $\mathbf{r} \in \mathbb{R}^n$. For sufficiently large $\alpha$, the Fourier transform exists. One well-known example in physics is the case $\alpha = n-2$, which is the Coulomb potential; its Fourier transform is $1/k^2$. For a general $\alpha$, I can argue that $$\widetilde{f}(\mathbf{k}) \sim \frac{1}{k^{n-\alpha}}$$ […]

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert x\rvert\rightarrow\infty$$ and $u$ is bounded as $$y\rightarrow\infty$$ I’ve tried hard for this,but my answer is not matching.I’ve gone through by applying infinite fourier transformation as my text book suggests,but I’m not even getting […]

Let $f, g\in L^{2},$ by Plancherel’s theorem, we have $$\langle f, g \rangle= \langle \hat{f}, \hat{g} \rangle.$$ My Question is: Is it true that: $$\langle f, g \rangle= \langle \hat{f}, \hat{g} \rangle$$ for $f\in \mathcal{S’}(\mathbb R^d)$ (tempered distribution) and $g\in \mathcal{S}(\mathbb R^d)$(Schwartz space)? If yes, how to justify.

I’m interested in this following definite integral: $$ \int_0^\infty du \frac{\sin(\beta u)}{1+u^\alpha}, $$ where $\beta>0$ and $\alpha\geq1$. Is there any closed form for this integral? I would be fine if the answer involves special functions, it would just be nice to have the answer in closed form. I’m also interested in the behavior of this […]

I would like to know how to calculate the Fourier transform of $$e^{A\sin(x)}$$ where $A$ is a real positive constant.

I need to prove this: $$ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$$ So far, I believe I have to use the Fourier transform standard equation $$ \mathcal F(f(x))=\frac{1}{2\pi}\int_{-\infty}^\infty f(x)e^{-isx}\,dx $$

Don’t know how to answer this question. please give the answer if anyone knows.. I am using this equation \begin{align} \sqrt{\frac{2}{\pi}}\int^\infty_0 f(x) \cos (\omega x) \,dx \end{align}

I am able to derive the following equation by substituting the definition of a Fourier transform into it’s inverse. $$2\pi\delta(x-x’) = \int_{-\infty}^{\infty} e^{ik(x-x’)} dk$$ How do you prove that the Dirac Delta is equal to an integral of the exponential function? How do you prove the above equation is true?

Let $f(z)=(\tau+z)^{-k}$. I want to apply the Poisson summation formula, to get that : $$\sum_{n \in \mathbb{Z}} (\tau+n)^{-k} = \frac{(-2 \pi i)^{k}}{(k-1)!} \sum_{m=1}^{\infty} m^{k-1}e^{2 \pi i m \tau}. $$ But then how can I compute the Fourier transform? I was thinking in residues but I don’t know which contour will be easiest for the computation, […]

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